Press Release From the ESB About Potential Releases From the Poulaphouca Reservoir

Poulaphouca Lake. © Eugene Brennan

Water isn't released directly into the River Liffey south of Ballymore Eustace from the Poulaphouca reservoir — it's discharged indirectly from the Poulaphouca dam into a compensatory lake in Ballymore. This acts as a buffer, protecting the river from huge surges and flooding. Although it's not related to surges on the river, a surge tank adjacent to the Liffey bridge over the gorge on the N81 protects the pipes, or penstocks, that deliver water to the turbines in the generating house from hydraulic shock. The compensatory lake also creates a pressure head for the Golden Falls hydro station as well as providing an amenity for fishing and water skiing. "The Flood" on the River Liffey is caused when water is discharged from this lake through the Golden Falls dam. The Poulaphouca reservoir has been acting a buffer up until now, holding large quantities of water from recent rainfall. The lake is fed by both the River Liffey and Kings River, in addition to some minor streams. However, water levels have increased by almost 2 metres, requiring water to be released. Since the Golden Falls lake is also almost full, this means that large quantities water must also be released from the Golden Falls dam into the Liffey at Ballymore Eustace to allow room for water being discharged from the Poulaphouca lake.
In a recent press release by the ESB, quantities haven't been specified and they say "there may be a need to pass through some of these additional inflows over the coming days."

Flows haven't been higher than 33 m³/s (cubic metres per second) in recent years. However, according to the Kilcullen Diary, they were at 55 m³/s in 2000 when a controlled, 24-hour discharge was necessary. this caused floodind in Kilcullen and other regions downstream.

Forecasted flow for the day, with values at 00:00, 08:00, 18:00 and 24:00 from the Golden Falls dam is available here on the ESB hydrometric page. 

Golden Falls dam. © Eugene Brennan

 

Devils Tower, the Hill of Allen, the Valley Park and Close Encounters of the Third Kind

Devil's Tower in Wyoming. Ben Stephenson from Cleveland, OH, CC BY 2.0 , via Wikimedia Commons

If you were watching Close Encounters of the Third Kind this afternoon (I hadn't seen it for maybe 15 years), you would have seen the towering rock formation that's used as a plot device and features in the end of the movie when the huge spaceship appears and the aliens emerge. This is Devils Tower in Wyoming, classed as a butte and possibly a laccolithic formation, composed of igneous rock. A butte is an isolated hill with steep, often vertical sides and a flat top. Igneous rock began as molten rock or magma that originated from inside the Earth, and then cooled. Examples are granite (which forms the Wicklow Mountains) and basalt (The Giant's Causeway). A laccolith is a body of molten rock that pushed up through layers or strata of softer rock such as sandstone, but didn't emerge as a volcano. Over hundreds of millions of years, the softer rock eroded away to leave the harder and steep sided tube. In Ireland we have several features on the landscape that would originally have been volcanoes. They include the Hill of Allen (what's left of it) and Croghan Hill near Daingean in Offaly. As far as I know, Andesite, the hard rock from Roadstone's Allen quarry was used for the walls and stone landscaping around St Brigid's Well in the Valley Park. Maybe someone can clarify this?

Top 100 Cool Science Facts for Kids!

© Eugene Brennan

Why is the sky blue? What is air made of? How many stars are there?
World of wonder fun science facts that every child should know! This article covers space, nature, technology, engineering, elementary mathematica, chemistry, physics and biology.
Science is fascinating and tries to explain how everything in the world and outer space works. Science gives us the answers to questions like "What is electricity?" and "How does an aeroplane fly?" Read on and learn 100 more cool science facts! 

Super Human Strength and an Injury on Church Mountain

Fence on Church Mountain, Co. Wicklow. © Eugene Brennan

In 2019 I was making my way down a gully on Church Mountain near Hollywood with my bike slung over my shoulder. I had cycled half-way up on the trails at the back of the mountain and was going to go further, but it was getting late in the day and I decided it would be wise to head home before it got dark. The gully was peppered with lots of rocks, covered in wet, slippery moss and algae and difficult to navigate. I was distracted by an annoying off-lead dog belonging to walkers behind me which kept barking at me. I lost my concentration, slipped and started to fall backwards, my bike pulling me down (which is why it's never a good idea to carry a ladder over your shoulder on a stairs). In that split second, I knew I'd smash my head off the rocks, so I threw myself forwards and upwards and managed to tear one of my quad muscles because of the amount of force it exerted. I had visions of having to call out mountain rescue, but managed to limp down to the road and make my way home, cycling with one leg. The next day I found it hard to walk and it took me 5 weeks to get back to normal. I hadn't done a lot of damage but this is by way of an introduction to an episode of Science With Dr Karl podcast where we learn how under extreme conditions, our muscles can literally tear themselves from the bones. Under normal circumstances, the "firmware" in our brains as Dr Karl describes it, limits the forces they exert.

Terminal Velocity of a Human, Free Fall and Drag Force

We all know that when an object is released from a certain height, it starts to fall. This, of course, is due to gravity, or more specifically the gravitational force of attraction between the object and the Earth. The force of gravity causes the object to accelerate and increase in velocity as it falls downwards towards the Earth. In actuality, both the Earth and the object are mutually attracted to each other, and the Earth moves upwards at the same time. However, since it's so massive in comparison to a small object and the force is so small, its movement is minuscule. 

Gravity exerts a force on everything. © Eugene Brennan

Definitions of Quantities Used in Kinematics

Before we go any further, let's define some of the terms used in kinematics, which is an area of physics concerned with the motion of objects.

• Mass: the amount of matter in an object. The greater the mass of an object, the greater the amount of inertia it has and its reluctance to move.
• Speed: the rate of change of position of an object (how fast something moves).
• Velocity: speed in a given direction. Velocity is a vector quantity, which means it has a magnitude called speed and also a direction. In physics, we generally talk about velocity rather than speed.
• Force: a push or pull. A force causes an object with mass to accelerate.
• Acceleration: the rate at which velocity changes.
• Free Fall: when an object falls under the influence of gravity alone without other forces acting on it.

See my beginner's guide to mechanics for a more detailed understanding of the basics of forces and motion:

Newton's Laws of Motion and Understanding Force, Mass, Acceleration, Velocity, Friction, Power and Vectors

Does Velocity Keep Increasing When Something Falls?

If an object falls in a vacuum outside Earth's atmosphere, its velocity continues to increase because of the acceleration due to gravity. This is called free fall. However, if the object falls through the air (or another fluid such as water), this limits the maximum velocity it can reach.

Velocity increases due to gravity. © Eugene Brennan

Drag Force

When an object moves through a fluid, it experiences a force which opposes motion and tends to slow it down. This force is called drag. The fluid could be a liquid, such as water, or a mixture of gasses, such as air. If you put your hand out the window of a moving car or try to wade through water, you can feel this force.

What Affects Drag?

Drag increases on an object as it moves faster. In fact, it increases exponentially, which means if velocity doubles, drag increases four times and if velocity triples, drag goes up nine times and so on. Drag is also influenced by the shape or surface geometry of an object. Aircraft, trains, ships, road vehicles, bullets and missiles are streamlined to minimise drag, reduce energy consumption and decrease in velocity.

When an object is dropped in a vacuum, it free falls, acted on by gravity alone. However, if it is dropped within Earth's atmosphere, it experiences drag which slows it down.

The force of gravity acts downwards, and the drag force acts upwards.

A force called drag opposes the force of gravity. © Eugene Brennan

 

The supersonic passenger aircraft Concorde. Aircraft have slim, streamlined profiles to reduce drag. Eduard Marmet, CC BY SA 2.0 Unported via Wikimedia Commons

What Is Weight?

Mass is the amount of matter in a body, but in physics, mass and weight have very specific meanings. While the mass of an object is the same, irrespective of where it is located in the Universe, weight varies. Weight is the gravitational force between objects and equals mass multiplied by the acceleration due to gravity g.

So the force of gravity or weight is

F_g = mg

Where F_g is the force measured in Newtons (N)

m is the mass of an object in kilograms (kg)

and g is the acceleration due to gravity in metres per second squared (m/s²)

g is approximately 9.81 metres per second per second, written as 9.81 m/s/s or m/s² 

There is a mutual attraction between Earth and other objects. © Eugene Brennan

At Equilibrium, Drag Force Equals Weight of the Object

The net force acting on a free-falling body is the difference between the weight acting down and the drag force acting upwards. As long as this is positive, the body keeps accelerating downwards.

Since the drag force increases with velocity, eventually at some stage it equals the weight of the falling body (which isn't changing and stays constant at F_g = mg).
Once this equilibrium point is reached, since the two forces are equal, there is no net force on the object. No net force means no more force to keep accelerating the body, so its velocity reaches a maximum known as the terminal velocity.

As velocity increases, the drag force acting upwards eventually equals the force of gravity acting downwards, the net force becomes zero and an object no longer accelerates. It has reached terminal velocity. © Eugene Brennan

Terminal Velocity of an Object

Terminal velocity is the maximum velocity reached by an object as it falls through a fluid.

Velocity of a Falling Object With No Drag

As an aside, let's look at the equation for the velocity of a falling object when there's no drag. If an object falls through a vacuum without being slowed down by a drag force, its velocity v in m/s is given by the equation:

v = √(2gh)

where g is the acceleration due to gravity.

and h is the distance fallen in metres (m)

In terms of time t in seconds since the object was dropped, another equation for velocity is:

v = gt

To put this into perspective, after 10 seconds of free fall in a vacuum, an object would be traveling at:

v = gt = 9.81 x 10 = 98.1 m/s or 355 km/hr (219 miles per hour)

However, as we shall see, drag puts an upper limit on velocity.

Without an atmosphere and drag, falling objects would increase in velocity until they hit the ground. © Eugene Brennan

The Drag Equation

The drag equation describes the force experienced by an object moving through a fluid:

If F_d is the drag force, then:

F_d = ½ ρ u² C_d A

Where F_d is the force in newtons (N)

p is the density of the fluid in kilograms per cubic metre (kg/m³)

u is the velocity of the object relative to the fluid in metres per second (m/s)

Cd is the drag coefficient that depends on the shape of the object and the nature of its surface

and A is the area of the orthogonal projection of the object in m². This can be visualised as the area of the shadow of the object cast on a surface if a light with a parallel beam was shone on it and landed perpendicular to the surface.
Because of the u² term in the equation, drag increases with the square of the velocity.

The drag equation

Drag coefficients. TheOtherJesse, public domain image via Wikimedia Commons

Derivation of Terminal Velocity

At equilibrium, the drag force F_d acting upwards equals the weight F_g
acting downwards

We know F_d = ½ ρ u² C_d A

and F_g = mg

At equilibrium, the velocity becomes the terminal velocity. Let's call it V_t

Equate F_g to F_d and replace u by V_t giving:

mg = ½ ρ u² C_d A = ½ ρ V_t² C_d A

So:

2mg = ρ V_t² C_d A

Divide both sides by ρ C_d A giving:

2mg / ρ C_d A = V_t²

Taking the square root of both sides gives us:

V_t = √((2mg) / (ρAC_d)

The terminal velocity equation

 

Terminal Velocity of a Human

From the equation for terminal velocity, we see it depends on several factors:

• Density of the air.
• Mass of the object
• Area of the object
• Acceleration due to gravity (this doesn't really change, so it can be assumed to be practically constant)
• The shape of the object

For a human, the drag coefficient C_d is about 1 in a belly down, horizontal orientation and 0.7 in head down position.

Typically in this position, terminal velocity is about 120 mph or 54 m/s.

Instantaneous and terminal velocity for a 100kg, 1.8m tall human lying horizontally. Terminal velocity is reached after about 14 seconds. © Eugene Brennan

 

How Long Does It Take to Reach Terminal Velocity and How Far Does a Human Fall?

It takes about 12 seconds to reach 97% of terminal velocity. During that period, a human would fall about 455 metres.

What Increases Terminal Velocity?

Speed skydivers compete by trying to reach the highest possible terminal velocity. From the equation, we can see that it can be increased by:

• being heavier
• diving in thinner, low density air
• reducing the projected area by diving head first
• reducing the drag coefficient by diving head first.
• wearing clothing that improves streamlining and reduces drag

Skydivers. Skeeze, Public domain image via Pixabay.com

 

References

Hannah, J. and Hillerr, M. J., (1971) Applied Mechanics (First metric ed. 1971) Pitman Books Ltd., London, England.

This article is accurate and true to the best of the author’s knowledge. Content is for informational or entertainment purposes only and does not substitute for personal counsel or professional advice in business, financial, legal, or technical matters.

© 2019 Eugene Brennan

How to Calculate the Sides and Angles of Triangles Using Pythagoras' Theorem, Sine and Cosine Rule

In this tutorial, we'll first learn the absolute basics about triangles. Then we'll learn about trigonometry which is a branch of mathematics that covers the relationship between the sides and angles of triangles.

What's Covered in the Tutorial?

• Polygons and the Definition of a Triangle
• The Basic Facts About Triangles
• The Triangle Inequality Theorem
• Different Types of Triangles
• Using the Greek Alphabet for Equations
• Sine, Cosine and Tangent
• Pythagoras's Theorem
• The Sine and Cosine Rules
• How to Work Out the Sides and Angles of a Triangle
• Measuring Angles
• How to Calculate the Area of a Triangle

 

What Is a Triangle?

By definition, a triangle is a polygon with three sides.

Polygons are plane shapes with several straight sides. "Plane" just means they're flat and two-dimensional. Other examples of polygons include squares, pentagons, hexagons and octagons. The word plane originates from the Greek polús meaning "many" and gōnía meaning "corner" or "angle." So polygon means "many corners." A triangle is the simplest possible polygon, having only three sides.

The Symbol for Degrees

Degrees can be written using the symbol º. So, 45º means 45 degrees.

Angles of a triangle range from greater than 0 to less than 180 degrees. © Eugene Brennan

Basic Facts About Triangles

• A triangle is a polygon with three sides.
• All the internal angles add up to a total of 180 degrees.
• The angle between two sides can be anything from greater than 0 to less than 180 degrees.
• The angle between two sides can't be 0 or 180 degrees, because the triangle would then become three straight lines superimposed on each other (These are called degenerate triangles).
• Similar triangles have the same angles, but different length sides.

 

No matter what the shape or size of a triangle, the sum of the 3 internal angles is 180. © Eugene Brennan

Similar triangles have the same angles but different length sides. © Eugene Brennan

 

What Are the Different Types of Triangles?

Before we learn how to work out the sides and angles of a triangle, it's important to know the names of the different types of triangles. The classification of a triangle depends on two factors:

• The length of a triangle's sides
• The angles of a triangle's corners

 

Type of Triangle by Lengths of Sides	Description
Isosceles	An isosceles triangle has two sides of equal length, and one side that is either longer or shorter than the equal sides.
Equilateral	All sides and angles are equal in length and degree.
Scalene	All sides and angles are of different lengths and degrees.

Types of triangles by length of sides.

 

Type of Triangle by Internal Angle	Description
Right (right angled)	One angle is 90 degrees.
Acute	Each of the three angles measures less than 90 degrees.
Obtuse	One angle is greater than 90 degrees.

 Types of triangles by angle.

Triangles classified by side and angles. © Eugene Brennan

What is Trigonometry?

Trigonometry is branch of mathematics that covers the relationship between the sides and angles of triangles. If we don't know all the sides or angles, we can use trigonometry to work out the unknown quantities.

How Do You Find the Sides and Angles of a Triangle?

There are several methods that can be used to calculate the unknown sides or angles. We can use formulas, mathematical rules, or the fact that the angles of all triangles add up to 180 degrees.

Tools to discover the sides and angles of a triangle

• Pythagoras's theorem
• Sine rule
• Cosine rule
• The fact that all angles add up to 180 degrees

Pythagoras's Theorem (The Pythagorean Theorem)

Pythagoras's theorem uses trigonometry to calculate the longest side (hypotenuse) of a right triangle (right angled triangle in British English). It states that for a right triangle:

The square on the hypotenuse equals the sum of the squares on the other two sides.

Pythagoras's Theorem. The hypotenuse is the longest side, opposite the right angle that measures 90 degrees. © Eugene Brennan

In the diagram above, the hypotenuse is the longest side of the right triangle, and is located opposite the right angle which is the angle that measures 90 degrees. So, if you know the lengths of two sides, all you have to do is square the two lengths, add the result, then take the square root of the sum to get the length of the hypotenuse.

If the sides of a triangle are a, b and c and c is the hypotenuse, Pythagoras's Theorem states that:

c² = a² + b² 

So

c = √(a² + b²)

 

Example Problem Using the Pythagorean Theorem

The sides of a triangle are 3 and 4 units long. What is the length of the hypotenuse?

Call the sides a, b, and c.
Side c is the hypotenuse.

a = 3
b = 4
c = Unknown

So, according to the Pythagorean theorem:

c² = a² + b² 

So

c² = 3² + 4² = 9 + 16 = 25

So c² = 25 and to find c, we just take the square root of 25 giving:

c = √25= 5

What are the Sine, Cosine and Tangent of an Angle?

A right triangle has one angle measuring 90 degrees. The side opposite this angle is known as the hypotenuse (another name for the longest side of this type of triangle). The length of the hypotenuse can be discovered using Pythagoras's theorem, but to discover the other two sides, sine and cosine must be used. These are trigonometric functions of an angle.

In the diagram below, one of the angles is represented by the Greek letter θ. (pronounced "thee - taa" ).

Sine, cosine and tan. © Eugene Brennan

 

Side a is known as the "opposite" side.
Side b is called the "adjacent" side.

The vertical lines "||" around the words below mean "length of."

So sine, cosine and tangent are defined as follows:

sine θ = |opposite side| / |hypotenuse| 

cosine θ = |adjacent side| / |hypotenuse|

tan θ = |opposite side| / |adjacent side|

Sine and cosine apply to an angle, not just an angle in a triangle, so it's possible to have two lines meeting at a point and to evaluate sine or cosine for that angle even though there's no triangle as such. However, sine and cosine are derived from the sides of an imaginary right triangle superimposed on the lines.
For instance, in the second diagram above, the purple triangle is scalene not right angled. However, you can imagine a right-angled triangle superimposed on the purple triangle, from which the opposite, adjacent and hypotenuse sides can be determined.
Over a range 0 to 90 degrees, sine ranges from 0 to 1, and cosine ranges from 1 to 0.
Remember, sine and cosine only depend on the angle, not the size of the triangle. So if the length a changes in the diagram above when the triangle changes in size, the hypotenuse c also changes in size, but the ratio of a to c remains constant. They are similar triangles.
Sine, cosine and tangent are often abbreviated to sin, cos and tan respectively.

The Sine Rule

The ratio of the length of a side of a triangle to the sine of the angle opposite is constant for all three sides and angles.

So, in the diagram below:

a / sine A = b / sine B = c / sine C

The sine rule. © Eugene Brennan

Now, you can check the sine of an angle using a scientific calculator or look it up online. In the old days before scientific calculators, we had to look up the value of the sine or cos of an angle in a book of tables.

The opposite or reverse function of sine is arcsine or "inverse sine", sometimes written as sin^-1. When you check the arcsine of a value, you're working out the angle which produced that value when the sine function was operated on it. So:

sin (30º) = 0.5 and sin^-1(0.5) = 30º

When should the sine rule be used?

The length of one side and the magnitude of the angle opposite is known. Then, if any of the other remaining angles or sides are known, all the angles and sides can be worked out.

Example showing how to use the sine rule to calculate the unknown side c. © Eugene Brennan

The Cosine Rule

For a triangle with sides a, b, and c, if a and b are known and C is the included angle (the angle between the sides), C can be worked out with the cosine rule. The formula is as follows:

c = a² + b² - 2ab cos C

 

The cosine rule. © Eugene Brennan

When should the cosine rule be used?

1. You know the lengths of the two sides of a triangle and the included angle. You can then work out the length of the remaining side using the cosine rule.
2. You know the lengths of all the sides but none of the angles. Rearranging the cosine rule equation gives the length of one of the sides.

c = a² + b² - 2ab cos C

Rearranging the equation:

C = arccos ((a² + b² - c²) / 2ab)

The other angles can be worked out similarly.

Example using the cosine rule. © Eugene Brennan

How to Find the Angles of a Triangle Knowing the Ratio of the Side Lengths

If you know the ratio of the side lengths, you can use the cosine rule to work out two angles then the remaining angle can be found knowing all angles add to 180 degrees.

Example:

A triangle has sides in the ratio 5:7:8. Find the angles.

Answer:

So call the sides a, b and c and the angles A, B and C and assume the sides are a = 5 units, b = 7 units and c = 8 units. It doesn't matter what the actual lengths of the sides are because all similar triangles have the same angles. So if we work out the values of the angles for a triangle which has a side a = 5 units, it gives us the result for all these similar triangles.

Use the cosine rule. So c² = a² + b² - 2ab cos C

Substitute for a,b and c giving:

8² = 5² + 7² - 2(5)(7) cos C

Working this out gives:

64 = 25 + 49 - 70 cos C

Simplifying and rearranging:

cos C = 1/7 and C = arccos(1/7).

You can use the cosine rule again or sine rule to find a second angle and the third angle can be found knowing all the angles add to 180 degrees.

Summary

If you've made it this far, you've learned numerous helpful methods to discover different aspects of a triangle. With all this information, you may be confused as to when you should use which method. The table below should help you identify which rule to use depending on the parameters you have been given.

Find the Angles and Sides of a Triangle: Which Rule Do I Use?

 

Known Parameters	Triangle Type	Rule to Use
Triangle is right and I know length of two sides.	SSS after Pythagoras's Theorem used	Use Pythagoras's Theorem to work out remaining side and sine rule to work out angles.
Triangle is right and I know the length of one side and one angle	AAS after third angle worked out	Use the trigonometric identities sine and cosine to work out the other sides and sum of angles (180 degrees) to work out remaining angle.
I know the length of two sides and the angle between them.	SAS	Use the cosine rule to work out remaining side and sine rule to work out remaining angles.
I know the length of two sides and the angle opposite one of them.	SSA	Use the sine rule to work out remaining angles and side.
I know the length of one side and all three angles.	AAS	Use the sine rule to work out the remaining sides.
I know the lengths of all three sides	SSS	Use the cosine rule in reverse to work out each angle. C = Arccos ((a² + b² - c²) / 2ab)
I know the length of a side and the angle at each end	AAS	Sum of three angles is 180 degrees so remaining angle can be calculated. Use the sine rule to work out the two unknown sides
I know the length of a side and one angle		You need to know more information, either one other side or one other angle. The exception is if the known angle is in a right angled triangle and not the right angle.

A summary of how to work out angles and sides of a triangle.

How to Get the Area of a Triangle

There are three methods that can be used to discover the area of a triangle.

Method 1. Using the perpendicular height

The area of a triangle can be determined by multiplying half the length of its base by the perpendicular height. Perpendicular means at right angles. But which side is the base? Well, you can use any of the three sides. Using a pencil, you can work out the area by drawing a perpendicular line from one side to the opposite corner using a set square, T-square, or protractor (or a carpenter's square if you're constructing something). Then, measure the length of the line and use the following formula to get the area:

Area = 1/2ah

"a" represents the length of the base of the triangle and "h" represents the height of the perpendicular line.

Working out the area of a triangle from the base length and perpendicular height. © Eugene Brennan

Method 2. Using side lengths and angles

The simple method above requires you to actually measure the height of a triangle. If you know the length of two of the sides and the included angle, you can work out the area analytically using sine and cosine (see diagram below).

Working out the area of a triangle from the lengths of two sides and the sine of the included angle. © Eugene Brennan

Method 3. Use Heron's formula

All you need to know are the lengths of the three sides.

Area = √(s(s - a)(s - b)(s - c))

Where s is the semiperimeter of the triangle

s = (a + b + c)/2

Using Heron's formula to work out the area of a triangle. © Eugene Brennan

Using the Greek Alphabet for Equations

In science, mathematics, and engineering many of the 24 characters of the Greek alphabet are borrowed for use in diagrams and for describing certain quantities.
You may have seen the character μ (mu) represent micro as in micrograms μg or micrometers μm. The capital letter Ω (omega) is the symbol for ohms in electrical engineering. λ (lambda) is used for wavelength, and, of course, π (pi) is the ratio of the circumference to the diameter of a circle.
In trigonometry, the characters θ (theta), φ (phi) and some others are often used for representing angles.

Letters of the Greek alphabet. © Eugene Brennan

How Do You Measure Angles?

Digital finder. Image courtesy Amazon

You can use a protractor or a digital angle finder like the one above from Amazon.These are useful for DIY and construction if you need to measure an angle between two sides, or transfer the angle to another object. You can use an angle finder as a replacement for a bevel gauge for transferring angles e.g. when marking the ends of rafters before cutting. Accuracy is usually down to 0.1 degrees.

You can draw and measure angles with a protractor. © Eugene Brennan

Triangles in the Real World

A triangle is the most basic polygon and can't be pushed out of shape easily, unlike a square. If you look closely, triangles are used in the designs of many machines and structures because the shape is so strong.

The strength of the triangle lies in the fact that when any of the corners are carrying weight, the side opposite acts as a tie, undergoing tension and preventing the framework from deforming. For example, on a roof truss, the horizontal ties (which can be joists in a ceiling) provide strength and prevent the roof from spreading out at the eaves.

The sides of a triangle can also act as struts, but in this case, they undergo compression. An example is a shelf bracket or the struts on the underside of a light aircraft wing or the tail wing itself.

Truss bridge. Image courtesy Kanenori on Pixabay

How to Implement the Cosine Rule in Excel

You can implement the cosine rule in Excel using the ACOS Excel function to evaluate arccos. This allows the included angle to be worked out, knowing all three sides of a triangle.

Calculating side lengths in Excel using the cosine rule. © Eugene Brennan

FAQs About Triangles

Below are some frequently asked questions about triangles.

What do the angles of a triangle add up to?

The interior angles of all triangles add up to 180 degrees.

What Is the hypotenuse of a triangle?

The hypotenuse of a triangle is its longest side.

What do the sides of a triangle add up to?

The sum of the sides of a triangle depends on the individual lengths of each side. Unlike the interior angles of a triangle, which always add up to 180 degrees

How do you calculate the area of a triangle?

To calculate the area of a triangle, simply use the formula:

Area = 1/2ah

"a" represents the length of the base of the triangle. "h" represents its height, which is discovered by drawing a perpendicular line from the base to the peak of the triangle.

How do you find the third side of a triangle that Is not right?

If you know two sides and the angle between them, use the cosine rule and plug in the values for the sides b, c, and the angle A.

Next, solve for side a.

Then use the angle value and the sine rule to solve for angle B.

Finally, use your knowledge that the angles of all triangles add up to 180 degrees to find angle C.

How do you find the missing side of a right angled triangle?

Use the Pythagorean theorem to find the missing side of a triangle. The formula is as follows:

c² = a² + b²

c = √(a² + b²)

What is the name of a triangle with two equal sides?

A triangle with two equal sides and one side that is longer or shorter than the others is called an isosceles triangle.

What is the cosine formula?

This formula gives the square on a side opposite an angle, knowing the angle between the other two known sides. For a triangle, with sides a, b and c and angles A, B and C the three formulas are:

a² = b² + c² - 2bc cos A

or

b² = a² + c² - 2ac cos B

or

c² = a² + b² - 2ab cos C

How to figure out the sides of a triangle if I know all the angles?

You need to know at least one side, otherwise, you can't work out the lengths of the triangle. There's no unique triangle that has all angles the same. Triangles with the same angles are similar but the ratio of sides for any two triangles is the same.

How to work out the sides of a triangle if I know all the sides?

Use the cosine rule in reverse.
The cosine rule states:

c² = a² + b² - 2ab cos C

Then, by rearranging the cosine rule equation, you can work out the angle

C = arccos ((a² + b² - c²) / 2ab)and

B

= arccos ((a²+ c² - b²) / 2ac)

The third angle A is (180 - C - B)

How to find the perimeter of a triangle

Finding the perimeter of a triangle is a straightforward operation. The perimeter is equivalent to the added lengths of all three sides.

perimeter = a + b + c

How to find the height of a triangle

Finding the height of a triangle is easy if you have the triangle's area. If you're given the area of the triangle:

height = 2 x area / base

If you don't have the area, but only have the side lengths of the triangle, use the following:

height = 0.5 x √ ((a + b + c)(-a + b + c)(a - b + c)(a + b - c)) / b

If you only have two sides and the angle between them, try this formula:

area = 0.5 (a)(b)(sin(γ)), then

height = area(sin(γ))

References

1. Trigonometry. from Wolfram MathWorld. (n.d.). Retrieved May 24, 2022, from https://mathworld.wolfram.com/topics/Trigonometry.html

2. Equilateral triangle. from Wolfram MathWorld. (n.d.). Retrieved May 24, 2022, from https://mathworld.wolfram.com/EquilateralTriangle.html

3. Isosceles Triangle. from Wolfram MathWorld. (n.d.). Retrieved May 24, 2022, from https://mathworld.wolfram.com/IsoscelesTriangle.html

4. Scalene Triangle. from Wolfram MathWorld. (n.d.). Retrieved May 24, 2022, from https://mathworld.wolfram.com/ScaleneTriangle.html

5. Prof. David E. Joyce. The laws of cosines and Sines. Laws of Cosines & Sines. (n.d.). Retrieved May 24, 2022, from https://www2.clarku.edu/faculty/djoyce/trig/laws.html

Disclaimer

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualised advice from a qualified professional.

© 2016 Eugene Brennan